Use binomial expansion to solve each exercise. (a) Find the coefficient of xy19 in (3x − 2y) 20.(b) Give a formula for the coefficient of x k in (x+1/ x) n .

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  (a) -29,884,416

  (b) C(n,(n -k)/2) . . . where C(n,k) = n!/(k!(n-k)!) and k ∈ ℕ

Step-by-step explanation:

(a) The coefficient of the k-th term of the expansion is ...

  C(20,k)(3x)^(20-k)(-2y)^k

For k=19, this is ...

  19(3x)(-2y)^19 = -29,884,416xy^19

The coefficient of the term is -29,884,416.

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(b) The p-th term of the expansion of (x +1/x)^n is ...

  C(n,p)(x^(n-p)(1/x)^p = C(n,p)(x^(n-2p))

For some exponent k of x, the value of p will be ...

  k = n -2p

  p = (n -k)/2

Then the coefficient of x^k will be C(n, (n-k)/2), where C(n, x) = n!/(x!(n-x)!).

You will notice that there will only be a term x^k for k having the same parity as n.

option 2 is correct.

step-by-step explanation:

we are given two inequality equation

x+y> 5 and -3x+2y< -2

first we draw the graph of both function using t-table method

equation: x+y> 5

for x=0, y=5

for x=-5, y=0

table:

x   :   0     5

y   :   5     0

test point (0,0) , 0> 5

false

equation: -3x+2y< -2

for x=0, y=-1

for x=2, y=4

table:

x   :   0     2

y   : -1       4

test point (0,0) , 0< -2

true

now we draw the graph. see the attachment for graph.

common region would be solution set.

option 2 is correct.